arXiv:quant-ph/0207087 v1 16 Jul 2002
Giovanna Morigi1∗, Sonja Franke-Arnold2, and Gian-Luca Oppo2 1
Max-Planck-Institut f¨ur Quantenoptik, Hans-Kopfermannstr. 1, D-85748 Garching, Germany
2
Dept. of Physics and Applied Physics, University of Strathclyde, 107 Rottenrow, Glasgow, G4 0NG United Kingdom
(Dated: June 28, 2006)
We study a four-level atomic scheme interacting with four lasers in a closed-loop configuration with a♦(diamond) geometry. We investigate the influence of the laser phases on the steady state. We show that, depending on the phases and the decay characteristic, the system can exhibit a variety of behaviors, including population inversion and complete depletion of an atomic state. We explain the phenomena in terms of multi-photon interference. We compare our results with the phase-dependent phenomena in the double-Λ scheme, as studied in [Korsunsky and Kosachiov, Phys. Rev A60, 4996 (1999)]. This investigation may be useful for developing non-linear optical devices, and for the spectroscopy and laser-cooling of alkali-earth atoms.
INTRODUCTION
Absorption and emission of monochromatic light in two-level atomic transitions are well-understood processes in quantum optics [1]. Their properties, however, can change drastically if transitions to a third atomic level have to be included. This is the case, for instance, in the Λ configuration where two stable states are coupled to a common excited state by laser fields, thus providing two excitation paths which can interfere. This interference lies at the heart of Coherent Population Trapping (CPT) [2]. Here, destructive interference between the transition amplitudes gives rise to a superposition of atomic states (dark state) that is decoupled from coherent radiation but populated by spontaneous emission. Consequently the atom becomes “trapped” in this coherent superposition.
For configurations like the Λ scheme, the relative phase of the laser fields does not affect the steady-state dynamics, in the sense that there always exists a reference frame in which the Rabi frequencies are real. This is no longer fulfilled in closed-loop configurations [3, 4], i.e. when a set of atomic states is (quasi- ) resonantly coupled by laser fields, such that each state of the set is connected to any other via two different paths of coherent photon-scattering: In this case, the relative phase Φ between the transitions determines the interference and hence critically influences the dynamics and steady state of the system [3, 4, 5]. Previous studies of closed-loop configurations often featured double-Λ systems, where two stable or metastable states are -each- coupled to two common excited states [5, 6, 7, 8]. These works have shown a rich variety of non-linear optical phenomena.
In this paper, we investigate the phase-dependent dynamics of a closed-loop configuration, consisting of four tran-sitions driven by lasers. One ground state is coupled in a V-type structure to two intermediate states, which are themselves coupled to a common excited state in a Λ-type structure. We label this system the♦(diamond) scheme. The steady-state of the♦scheme, like that of the double-Λ configuration, is a periodic function of the relative phase Φ between the excitation paths, which contribute to the scattering between any initial and final state of the scheme. In particular, the steady state is determined by the concurrence of the phase-dependent Hamiltonian dynamics and the relaxation processes. We will show that, depending on Φ and on the lifetimes of the intermediate states, the ♦
system can show a variety of behaviors including population inversion, CPT, and phase-dependent refractive indices. It is worth noting that the double-Λ and the♦schemes are governed by the same Hamiltonian, but are characterized by different relaxation processes. This results in critical differences in the dynamics, which we will point out in our discussion.
Excitation configurations like the ♦ scheme have been investigated in the literature as a model for observing pressure-induced resonances [9], and can be found for instance in experiments with gases of alkali-earth atoms which aim at optical frequency standards [10] or at reaching the quantum degeneracy regime by all-optical means [11]. Our investigation may contribute to the spectroscopy of these systems and to the development of new and efficient methods of laser-cooling.
THE MODEL
In this section we introduce the model, using a density-matrix formalism. We move then to a reference frame where the phase dependence is explicit, and formulate the optical Bloch equations. Finally, we discuss the symmetries of the system as a function of the phase and of the relaxation processes.
The Master equation for a single atom
We consider a gas of atoms of massm. The atoms are free, and interact with a multi-chromatic light field. For a sufficiently-dilute gas, each atom interacts individually with the light, which couples to a set of atomic levels as depicted in Fig. 1. This set contains a ground state|1i, two intermediate states|2iand|3iand an excited state|4i. The transitions|1i → |2i,|3iand|2i,|3i → |4iare optical dipoles with decay ratesγ2,γ3,γ42, andγ43, respectively. Each
dipole transition is driven resonantly by a laser, which is here considered to be a classical running wave, propagating along the ˆz axis. The laser coupling to the transition |ii → |ji is characterized by frequency ωij and wave vector
kij, while the strength of the coupling is given by the Rabi frequency gijeiχij, whereg
ij is real andχij is a constant phase, determined by the phase of the atomic dipole and by the phase of the laser at timet= 0 and positionz= 0. The state of one atom at timet is described by the density matrixσ, which obeys the Master equation:
∂
∂tσ=
1
i¯h[H(t), σ] +Lσ. (1)
Here, the HamiltonianH(t) contains the coherent dynamics of the atom and the LiouvillianLdescribes the relaxation processes. The Hamiltonian depends explicitly on time and reads:
H(t) = p
2
z 2m+
4
X
j=2
¯
hωj|jihj|
+¯h 2
X
j=2,3
g1je−i(ω1jt−k1jz+χ1j)|jih1|
+gj4e−i(ωj4t−kj4z+χj4)|4ihj|+ H.c.. (2)
The first term corresponds to the kinetic energy of the atomic center of mass, wherepz is its momentum along the ˆ
z axis (for simplicity, we consider only the component of the atomic motion along the ˆz axis). The second term corresponds to the internal Hamiltonian, where ¯hωj denote the energies of the atomic states|jirelative to the energy of the state|1i. The remaining term describes the atom-laser interaction.
The relaxation processes are assumed to be solely radiative, and the LiouvillianLin (1) has the form:
Lσ = X j=2,3
γ4j
Z 1
−1
duN4j(u)e−ikj4uz|jih4|σ|4ihj|eikj4uz−
γ4
2 (|4ih4|σ+σ|4ih4|) (3)
+ X
j=2,3
γj
Z 1
−1
duNj1(u)
e−ik1juz
|1ihj|σ|jih1|eik1juz
−12(|jihj|σ+σ|jihj|)
,
where γ4 = γ42+γ43 is the total decay rate from level |4i, and Nij(u) is the dipole pattern for the spontaneous
emission of a photon on the transition |ii → |ji (j = 2,3,4), i.e. the probability of emission at the angle θ from the ˆz axis, such thatu= cosθ. This term describes the diffusion in the dynamics of the atomic motion due to the randomness of the incoherent events. In the limit in which the center-of-mass motion can be treated classically, and for sufficiently short times this effect can be neglected so that the explicit form of the dipole pattern is irrelevant. On this time scale and for a dilute gas, we can neglect relaxations due to collisions between the atoms. This is the regime that we are considering in the main body of the paper.
Change of reference frame
configurations [4], and it is a manifestation of the intrinsic phase sensitivity of the dynamics. However, in an adequate reference frame, the Rabi frequencies can be chosen such that only one is complex, with its phase Φ being a function of all laser phases. Without loss of generality, we move to a reference frame where Φ is associated with the laser coupling to the transition|3i → |4i, so that the coherent dynamics of the system is now described by the Hamiltonian [3]
˜
H(Φ) = p
2
z 2m+
4
X
j=2
¯
hδj|jihj|
+¯h
2 g12|2ih1|+g13|3ih1|+g24|4ih2|+g34e
iΦ
|4ih3|+ H.c.
, (4)
where Φ = Φ(t, z). The new Hamiltonian is connected withH(t) by the relation: ˜H(Φ) =U−1H(t)U−i¯hU−1∂U/∂t, whereU(t) is a unitary transformation, which reads
U(t) = exp{−i [(ω12+ω24)t−(k12+k24)z+ (χ12+χ24)]|4ih4|} (5)
× exp{−i [(ω12t−k12z+χ12)|2ih2|+ (ω13t−k13z+χ13)|3ih3|]}.
The detuningsδj in (4) are given by
δ2 = (ω2−ω12) +k12pz
m +
¯
hk2
12
2m ,
δ3 = (ω3−ω13) +
k13pz
m +
¯
hk2
13
2m ,
δ4 = (ω4−ω12−ω24) +(k12+k24)pz
m +
¯
h(k12+k24)2
2m ,
and the phase Φ is defined as
Φ = ∆ωt−∆kz+ ∆χ, (6)
where
∆ω =. ω12+ω24−ω13−ω34, (7)
∆k =. k12+k24−k13−k34, (8)
∆χ =. χ12+χ24−χ13−χ34. (9)
The phase Φ is the relative phase between the two excitation paths characterizing any transition between two atomic states. This phase is in general time- and position-dependent, and it results from the multi-photon detuning ∆ω, the wave-vector mismatch ∆k, and from the initial laser and atomic-dipole phases, ∆χ.
In the new reference frame the density matrix is ρ = U−1(t)σU(t), and its evolution is described by the Master Equation
∂
∂tρ=
1 i¯h
h ˜
H(Φ), ρi+Lρ . (10)
In the next section we derive the equations of motion for the matrix elements ofρ.
Optical Bloch Equations
In the following we assume a thermal distribution of atoms at temperatureT, such that the thermal energykBT is much larger than the recoil energies ¯h2k2
ij/2m. In this limit, we treat the atomic motion classically, and neglect the effects of the photon recoil on the center-of-mass dynamics [12]. Then, the one-atom density matrix is given by
ρ= R
withi, j= 1,2,3,4 (we drop the explicit dependence on the parameters (z, pz) ), the Optical Bloch Equations (OBE) have the form:
˙
ρ11 = ig12
2 (ρ12−ρ21) + i
g13
2 (ρ13−ρ31)
+γ2ρ22+γ3ρ33, (11)
˙
ρ22 = −i
g12
2 (ρ12−ρ21) + i
g24
2 (ρ24−ρ42)
+γ42ρ44−γ2ρ22, (12)
˙
ρ33 = −i
g13
2 (ρ13−ρ31) + i
g34
2 (e
iΦρ
34−e−iΦρ43)
+γ43ρ44−γ3ρ33, (13)
ρ44 = 1−ρ11−ρ22−ρ33, (14)
˙
ρ12 =
iδ2−γ2
2
ρ12−ig12
2 (ρ22−ρ11)
−ig13 2 ρ32+ i
g24
2 ρ14, (15)
˙
ρ13 =
iδ3−
γ3
2
ρ13−i
g13
2 (ρ33−ρ11)
−ig12 2 ρ23+ i
g34
2 e
iΦρ
14, (16)
˙
ρ24 =
−i(δ2−δ4)−
γ2+γ4
2
ρ24−i
g24
2 (ρ44−ρ22)
−ig12 2 ρ14+ i
g34
2 e −iΦρ
23, (17)
˙
ρ34 =
−i(δ3−δ4)−
γ3+γ4
2
ρ34−i
g34
2 e −iΦ(ρ
44−ρ33)
−ig13 2 ρ14+ i
g24
2 ρ32, (18)
˙
ρ14 =
h iδ4−
γ4
2 i
ρ14−i
g12
2 ρ24−i
g13
2 ρ34 +ig24
2 ρ12+ i
g34
2 e −iΦρ
13, (19)
˙
ρ23 =
−i (δ2−δ3)−
γ2+γ3
2
ρ23−i
g12
2 ρ13
−ig24 2 ρ43+ i
g13
2 ρ21+ i
g34
2 e
iΦρ
24, (20)
andρji= (ρij)∗. In the following we will refer to the diagonal elements as “populations”, giving the occupation of the atomic states, and to the off-diagonal elements as “one-photon coherences” or “two-photon coherences”, depending on whether the involved states are coupled at lowest order by the scattering of one or two photons, respectively.
Discussion
have been obtained in the presence of relaxation processes.
An interesting manifestation of the phase-dependent dynamics is the probability of two-photon absorption on the transition|1i → |4i. In the limit of weak excitations it has the form
P1→4∝
g12g24
δ2−iγ2/2
+g13g34exp(iΦ)
δ3−iγ3/2
2
. (21)
Here, the first and second terms on the RHS describe the transitions via the intermediate states|2iand|3i, respectively. From (21) it is evident that both the phase difference Φ as well as the ratio of the laser detuningsδ2, δ3and the decay
ratesγ2, γ3 determine the interference between the two paths. In the special case of equal Rabi frequenciesgij =g, and forδ2=δ3, γ2=γ3, the role of Φ is singled out:
P1→4∝cos
2Φ
2 . (22)
Thus, the transition probability from the ground to the excited level is modulated by Φ. In particular, it is maximal for the values Φ = 2nπ(wherenis an integer), while it vanishes for Φ = (2n+ 1)π. At the latter value, no transition to|4ioccurs. In the next section we will show thatP1→4always vanishes for Φ = (2n+ 1)πat steady state, even when the system is driven at saturation. We remark that the appearance of this behavior requires a ”symmetric” excita-tion configuraexcita-tion, meaning that each two-photon excitaexcita-tion path from|1ito|4ihas, separately, the same probability. A further understanding of the problem can be gained by moving to a suitable basis, following the analysis of [3]. This basis is chosen appropriate to the structure of the Hamiltonian (4) and the relaxation processes in (3). For
gij =gandδj= 0 the dynamics offers simple interpretations for Φ =nπ.
We first focus on the values Φ = (2n + 1)π. Here, it is convenient to use the orthogonal basis set
{|1i,|4i,|Ψ23(0)i,|Ψ23(π)i}, where
|Ψ23(θ)i=
1
√
2
|2i+ e−iθ
|3i
(23) withθ= 0, π. In this basis, the Hamiltonian (4) can be rewritten as
˜
H((2n+ 1)π) = √¯hg
2 h
|Ψ23(0)ih1|+|4ihΨ23(π)|+ H.c.
i
, (24)
where we have omitted the atomic motion. Thus, (24) describes two-level dynamics within the orthogonal subspaces
{|1i,|Ψ23(0)i} and {|4i,|Ψ23(π)i}. These subspaces are coupled by spontaneous decay, and the coupling between
states due to coherent and incoherent processes is represented in Fig. 2 a). From the structure of the decay, it is evident that the atom is eventually pumped into{|1i,|Ψ23(0)i}. Hence, the steady state of the system for this value
of the phase corresponds to that of the driven two-level transition|1i → |Ψ23(0)i.
For Φ = 2nπwe describe the system in the orthogonal basis set{|Ψ14(0)i,|Ψ14(π)i,|Ψ23(0)i,|Ψ23(π)i}, where
|Ψ14(θ)i=
1
√
2
|1i+ eiθ
|4i
(25)
withθ= 0, π. In this basis, the Hamiltonian ˜H can be written as ˜
H(2nπ) = ¯hgh|Ψ23(0)ihΨ14(0)|+ H.c.
i
, (26)
and it describes a coherent two-level dynamics between the states |Ψ23(0)i and |Ψ14(0)i. The states |Ψ14(π)i and
|Ψ23(π)iare decoupled from the coherent drive because of destructive interference between the corresponding
excita-tion paths, and from this point of view they are dark states. However, they are not stable, and decay with ratesγ4
andγ2+γ3 respectively. The level scheme in the new basis is plotted in Fig. 2 b). Here, it is evident that the system
is incoherently pumped among the driven transition|Ψ23(0)i → |Ψ14(0)iand the two dark states. One could say that
the steady state is determined by the competition between the Hamiltonian dynamics and the relaxation processes. Thus, some localization in one of the dark superpositions (CPT) can occur, if this is more stable than the other, i.e. if the rate of pumping into it is much larger than its decay rate. In order to quantify this effect, we introduce the parameterαdefined as:
α= γ4
γ2+γ3
Thus, for Φ = 0 andα≫1 (α≪1) the dark state|Ψ23(π)i(|Ψ14(π)i) is long lived with respect to all other states and,
at steady state, it has a high probability of occupation. Such probability increases the moreαdiffers from unity, and approaches 1 forα→ ∞(α→0), corresponding to the system being trapped in|Ψ23(π)i(|Ψ14(π)i). We will show
that due to this effect, population inversion can occur on the transition|1i → |2i,|3iforα≫1 and on|2i,|3i → |4i
forα≪1. On the other hand, such behavior disappears asαapproaches 1. Forα= 1 and at saturation, the system is equally scattered among all states.
Comparison with the double-Λconfiguration
In the absence of spontaneous decay, the♦configuration is formally identical to the double-Λ scheme, extensively studied in the literature [4, 5, 6]. Thus, the symmetries induced on the coherent dynamics by the phase are exactly the same [3]. We have discussed, however, that the steady state is critically determined by the concurrence between this symmetry and the relaxation processes. Thus, the introduction of the spontaneous decay leads to critical differences between the two systems. For an easier comparison we are labelling the atomic states of the double-Λ system as shown in the inset of Fig. 1. In this scheme the excited states |1i and |4i decay spontaneously into the stable or metastable states |2i and |3i. In the ♦ scheme, the excited state |4i decays into the intermediate states |2i and
|3i, which themselves decay into the ground state |1i. A first difference is that in the ♦ scheme the dynamics will be phase-sensitive only when the V- or the Λ scheme (or both) are driven at saturation, while below saturation it will reduce to the well-known V-configuration. In the double-Λ system, on the other hand, phase-sensitive dynamics survives also well below saturation [6].
When looking at the behavior as a function of Φ, the differences are more striking: at Φ = (2n+ 1)π, for instance, the
♦scheme is pumped into the subspace{|1i,|Ψ23(0)i}, which is a closed two-level transition, for the coherent drive as
well as for the relaxation processes. In the double-Λ scheme, instead, the atom can be found in any of the four states due to incoherent coupling [14].
At Φ = 2nπ, the role of the dark states differs between the two configurations. In the double-Λ system CPT occurs in the state|Ψ23(π)i, which is completely dark and, in the absence of other sources of decay, stable. In this configuration,
and for equal decay rates from the excited states, the state|Ψ14(π)iis never accessed. In the♦scheme, instead, both
dark states are accessed, and (partial) CPT occurs only when their decay rates differ substantially from one another.
STEADY-STATE SOLUTIONS
In this section we study the steady-state solution of (10) as a function of the phase Φ. We consider laser frequencies and geometries that fulfill ∆k = 0 and ∆ω = 0, so that Φ does not depend on time and space. In order to obtain simple analytic solutions we consider resonant drives so thatδj = 0.Further, we assume that the moduli of the Rabi frequencies are all equal,gij =g, and that the decay rates fulfill the relation γ2 =γ3=γ,γ42=γ43 =γ4/2. Under
these assumptions, the system exhibits symmetry in the clockwise and anti-clockwise multiphoton excitation paths, the difference being the phase Φ.
In this limit, we report and discuss the steady-state solutions of the OBE in (11-20) as a function of the phase Φ and of the dimensionless parameters Ω =g/γ andα=γ4/(2γ), defined in (27). We remark that in the following the
rotated one-photon coherence
˜
ρ34=ρ34eiΦ (28)
is reported in the results. This frame allows to identify the real and imaginary parts of ρ12, ρ13, ρ24, ˜ρ34 with the
dispersive and absorptive response of the atomic medium to the fields which couple the corresponding transitions [15]. At the end of this section we will discuss experimental situations under which the assumptions given above are justified and discuss our results for generic values of δ, in relation to the assumption of classical motion, and for unequal Rabi frequencies and decay rates.
Caseα= 1
We first consider the case α = 1. For convenience, we separate the real and imaginary part of the coherences, denoting them with uij = Re{ρij}, vij = Im{ρij}, respectively (here, ˜u34 = Re{ρ˜34}, ˜v34 = Im{ρ˜34}). The
ρ(11ss) =
1
D
h 1 + 16
3 Ω
2+19
9 Ω
4
sin2Φ
2 + 3
+4 9Ω
6
5 sin2Φ
2 + 3
+2 9Ω
8sin2Φi, (29)
ρ(22ss)=ρ
(ss)
33 =
Ω2
D
h 1 +1
3Ω
2
7 + 3 sin2Φ 2 +4 9Ω 4
3 + sin2Φ 2
+2
9Ω
6sin2Φi, (30)
ρ(44ss) =
Ω4
D cos
2Φ
2
1 + 4 3Ω
2+8
9Ω
4sin2Φ
2
, (31)
u(12ss)=−u
(ss)
13 =
Ω3
2Dsin Φ
−1 +2 3Ω
2+8
9Ω
4sin2Φ
2
, (32)
v12(ss)=v
(ss)
13 =
Ω
D
h 1 + 1
3Ω
2
7 + 3 sin2Φ 2 +4 9Ω 4
3 + sin2Φ 2
+2
9Ω
6sin2Φi, (33)
u(24ss)=−u˜
(ss)
34 =
Ω3
2Dsin Φ
1 + 2Ω2+8 9Ω
4sin2Φ
2
, (34)
v24(ss)= ˜v34(ss) = Ω
3
D cos
2Φ
2
1 + 4 3Ω
2+8
9Ω
4sin2Φ
2
, (35)
u(14ss) = −
Ω2
D cos
2Φ
2
1 +4 3Ω
2+4
9Ω
4sin2Φ
2
, (36)
v14(ss) = Ω
2
2Dsin Φ
1 + 4
3Ω
2+4
9Ω
4sin2Φ
2
, (37)
u(23ss) =
Ω2
D
h 1 +2
3Ω
2
2 + 3 sin2Φ 2
+4 9Ω
4sin2Φ
2
1 + 3 sin2Φ 2
i
, (38)
v23(ss) = −
Ω4
D sin Φ
1 +2
3Ω
2sin2Φ
2
, (39)
where
D = 1 +22 3 Ω
2+4
9Ω
4
7 sin2Φ 2 + 27
+16 9 Ω
6sin2Φ
2 + 3
+8 9Ω
8sin2Φ. (40)
The form of the solutions allows to identify the contributions of the various multi-photon processes to the steady state. For instance, at second order in Ω (i.e. at second order ing/γ) onlyρ14depends on the phase while the populations,
one-photon coherences and ρ23 are independent of Φ, and ρ44, ρ24, ρ34, u12, u13, and v23 vanish. In fact, this limit
corresponds to weak drives, and the relevant processes consist in resonant scattering on the transitions|1i → |2i,|3i. Thus, at second order in Ω the system is equivalent to a V configuration driven below saturation.
At higher order, the steady-state solutions are phase dependent. This is evident, e.g., in the excited-state population, which is proportional to cos2(Φ/2). In particular, at lowest order in Ω,ρ
44≈Ω4cos2(Φ/2). However, as Ω is increased
this modulated dependence of the populations is lost: at leading order in Ω, and for Φ 6= (2n+ 1)π, all states are equally populated. An exceptional behavior occurs at Φ = (2n+ 1)π. Here, ρ44 = 0 at all orders, while at leading
order ρ11 = 2ρ22 = 2ρ33 = 1/2. In Fig. 3 b) the populations are plotted as a function of the phase for Ω = 2 and
α= 1. (For comparison, Figs. 3 a) and 3 c) plot the populations forα≪1 andα≫1, respectively; we will discuss these regimes in the next subsection.) For the chosen parameters, ρ11, ρ44 vary with Φ, while ρ22, ρ33 are almost
Fig. 3 e) and 3 h) show the one-photon coherences (32-35) as a function of Φ for Ω = 2 andα= 1. Their real parts vanish for Φ = nπ, and one can easily verify from (32-35) that this occurs at all orders in Ω. This is a feature of resonantly-driven two-level systems, and this result is consistent with the analysis of the previous section. Moreover, for Φ = (2n+ 1)πone finds ρ24=ρ34= 0, which is consistent with the vanishing ofρ44. It is worth noting thatu12
and u13 can show additional zeros, as can be seen from their analytic form. These zeros depend on the values of Φ
and Ω, which forα= 1 satisfy the relation sin2(Φ/2) = 3(3
−2Ω2)/8Ω4. Thus, they exist only for a certain range of
values of the Rabi frequency. Their existence can be interpreted as interference of multi-photon scattering at all orders. The two-photon coherenceρ14 is proportional to cos(Φ/2), a result in agreement with the calculation in (22). The
interpretation of the two-photon coherences becomes more transparent by employing the basis of the previous section. For instance,ρ14can be expressed as:
u14 =
1
2[hΨ14(0)|ρ|Ψ14(0)i − hΨ14(π)|ρ|Ψ14(π)i]
v14 = Im{hΨ14(π)|ρ|Ψ14(0)i}.
Analogue equations hold for u23 and v23. Thus, u14 =−1/2 (+1/2) corresponds to the system being in the state
|Ψ14(π)i (|Ψ14(0)i). The imaginary part v14 measures the coherence between these two states. We now look at
(36-39) as a function of Φ, which are plotted in Fig. 3 k) and 3 n) for Ω = 2. The behavior we observe is consistent with the above interpretation in the superposition basis, and with the discussion of the populations and one-photon coherences. At Φ = 2nπthe imaginary partv14 (v23) vanishes, supporting the hypothesis of no coherence between
|Ψ14(π)i and |Ψ14(0)i (|Ψ23(π)i and |Ψ23(0)i). Moreover, u14 =−u23 < 0, which implies, after a straightforward
calculation, that the probability to find the system in the dark states is 1/2. Thus, it is not proper to speak of CPT for these parameters.
At leading order in Ω the coherences vanish for Φ = 2nπ, in agreement with the expectation that at saturation the system is equally distributed between all states. At Φ = (2n+ 1)π one finds ρ14 = v23 = 0 while u23 is
posi-tive and exhibits a local maximum. This is consistent with the picture of two-level dynamics between|1iand|Ψ23(0)i.
So far we have discussed the case α = 1, when the relaxation rates of the two-photon coherences are the same. We have seen that the features of the phase-induced dynamics are always recognizable in the coherences. However, at steady state the atom is not localized in a particular atomic level or coherent superposition of atomic levels. In general, the dependence of the populations on the phase is washed out for increasing Rabi frequencies, except for the vanishing ofρ44at Φ = (2n+ 1)π. This is understood by considering that the steady state is given by the concurrence
of the coherent drive, which has a phase-dependent symmetry, and the relaxation processes with a fixed structure of the coupling between the atomic states. For any value of Φ6= (2n+ 1)π, the two effects compete, and at saturation a kind of ’ergodicity’ is recovered, so that the atomic states are equally populated. On the contrary, for Φ = (2n+ 1)π, an eigenspace of the coherent scattering processes exists and is preserved by the action of the incoherent processes. Consequently, at any value of Ω andαthe occupation of the state|4ivanishes.
Caseα6= 1
Figures 3 a)-o) plot the steady-state solutions of the OBE for α= 0.1,1,10. Comparing the curves, we see some general features in the behavior at different α. For instance, the population of the state |4i is always zero for Φ = (2n+ 1)π. This value of the phase is also a pole of the coherencesρ24, ρ34, ρ14, and of the real parts u12, u23.
Here, the population of the state|1i and the real part of the two-photon coherence u23 exhibit a local maximum.
These results are all consistent with the picture of two-level dynamics between|1iand |Ψ23(0)i, as discussed in the
previous section. The steady-state values have a very transparent form for Φ = (2n+ 1)π, and read:
ρss
11 =
1 + 2Ω2
1 + 4Ω2, (41)
ρss
22=ρss33 =
Ω2
1 + 4Ω2, (42)
ρss44 = 0, (43)
v12ss=v
ss
13 =
Ω
uss
23 =
Ω2
1 + 4Ω2, (45)
uss
12 = uss13=ρss24=ρ34ss=v23ss=ρss14= 0, (46)
which have been evaluated for Ω = g/γ and an arbitrary value of α=γ4/2γ. In these solutions, the parameterγ4
does not appear, showing once again that the level|4idoes not affect the steady-state dynamics for this value of the phase.
A striking difference among the three regimes appears at values of the phase close to Φ = 2nπ. Here, we find population inversion on the transitions|1i → |2i,|3iforα= 10, (|2i,|3i → |4iforα= 0.1), while the real part of the two-photon coherenceu23(u14) approaches the value−1/2. At this value of Φ, we write the steady-state solutions as
a function ofγ,α=γ4/2γand Ω =g/γ:
ρss
11 =
1
D[α
2(1 + 2α) + Ω2α(3 + 5α+ 4α2) + Ω4(1 + 2α)], (47)
ρss
22=ρss33 =
αΩ2
D [α(1 + 2α) + Ω
2(α+ 2)], (48)
ρss
44 =
Ω4
D(1 + 2α), (49)
vss
12=vss13 =
αΩ
D
α(1 + 2α) + Ω2(2 +α)
(50)
vss
24= ˜vss34 =
αΩ3
D (1 + 2α) (51)
uss
23 =
αΩ2
D [α(1 + 2α) + Ω
2(1
−α)] (52)
uss
14 = −
Ω2
D[α(1 + 2α) + Ω
2(1
−α)] (53)
uss
12 = uss13=v14ss=v23ss=uss24= ˜uss34= 0, (54)
where
D=α2(1 + 2α) + Ω2α(3 + 7α+ 8α2) + 2Ω4(1 + 4α+α2). (55)
These results are plotted as a function ofαin Fig. 4, by keeping γand Ω as fixed parameters.
Here, we see that forα≪1 the system is localized in the atomic states|1i,|4i, and the coherenceu14has a maximum
absolute value. In particular, for α → 0 the populations of the states |2i,|3i vanish together with the imaginary part of all coherences. In this case the atom is in the dark state |Ψ14(π)i, which is stable, and CPT occurs. Such
localization persists for small values ofα, although the populations of the intermediate states – and the incoherent scattering processes – increase asαapproaches 1. It is interesting that for these (small) values ofαthe system exhibits population inversion on the transitions|2i,|3i → |4i. Analogously, it can be verified that, fixed γ4 andg, for γ→0
the system is trapped in the dark state|Ψ23(π)i: CPT occurs in this coherence, and this implies population inversion
on the transition|1i → |2i,|3i. Note that the localization in an atomic superposition persists in the neighbourhood of the value of the phase Φ = 2nπ, as it is visible in Fig. 3 a) and c). For instance, forα≫1 population inversion occurs on the transition|2i,|3i → |1ion an interval of values [2nπ−Φ0,2nπ+ Φ0]. The phase Φ0satisfies the relation
ρ22(Φ0), ρ33(Φ0) =ρ11(Φ0), and in general Φ0 can be said to separate two regimes, where the dynamics associated
with the symmetry at phase 2nπor with Φ = (2n+ 1)πprevails.
It is interesting to note that for Φ = 2nπthe populations and in particular the decay dependent population inversion show trends typical of a three-level cascade system while for Φ = (2n+ 1)π the system is effectively reduced to a V-configuration because ofρ44= 0.
Finally, we emphasize the additional poles of the one-photon coherences which we have identified in the analytical solutions for α= 1. We have interpreted their origin as due to photon scattering at all orders. We remark that they appear inu12andu13forα= 0.1, see Fig. 3 d), and in u24 and ˜u34 forα= 10, see Fig. 3 f).
Discussion
Γ = min(γ, γ4). Provided that the linewidth Γ is much larger than the recoil energies, Γ≫¯h2kij2/2m, the presented results describe sensibly the atomic response to the drive.
When the medium is Doppler broadened, i.e. forκBT >¯hΓ/2 (still keeping the constraint on the recoil energies), many features discussed for the case pz = 0 survive and will appear in the signal measured over the ensemble, provided thatδ2=δ3=δ δ4= 0, so that two-photon transitions are Doppler-free. This situation can be realised for
degenerate intermediate-state energies (ω2=ω3), resonant drives (ω12 =ω13 =ω2−ω1,ω24 =ω34 =ω4−ω2), and
laser geometries such that the wave vectors fulfill the relation k12 =k13 ∼ −k24 = −k34 =k. In this way Φ does
not depend on time and space (∆ω = 0, ∆k = 0), and δ is given by kpz/m. Also in this regime we find that for Φ = (2n+ 1)πthe population of|4ivanishes independently ofδ, together with the coherences ρ24, ρ34 andρ14. For
Φ = 2nπandαsufficiently different from unity, population inversion can be observed provided the atomic transitions are saturated [16].
Finally, we remark that only part of these considerations can be applicable to ”asymmetric” configurations, i.e. for values of the Rabi frequencies, eigenenergies, decay rates, etc. , which change the structure of the Hamiltonian and relaxation processes, introducing thus either different weights to the interfering excitation paths, and/or additional relative phases, and/or different resonances. Here, the dependence of the steady state on the phase Φ cannot often be simply singled out. The dynamics is a complex combination of all parameters, and exhibits an extremely rich variety of phenomena, which will be subject of future investigations.
CONCLUSION
We have studied the dynamics of a 4-level system interacting with lasers in a configuration which we have labelled the♦scheme because of its geometry. This scheme has a closed-loop excitation structure [3, 4], i.e. any transition amplitude between two given states is the sum of two contributions, corresponding to two excitation paths, which may interfere. The dynamics is determined by a large number of parameters. Here, we have considered that both paths have the same weight, while they differ by a relative phase Φ. We have discussed the origin of Φ, and investigated the steady state of the interacting system as a function of Φ, in the regime where the steady-state solution exists.
For the chosen parameters, the steady-state solution is phase-sensitive. This is particularly evident in the coherences, whereas in general the phase dependence of the population is particularly enhanced for certain ranges of values of the relaxation rates. In particular, when the lifetimes of the intermediate states are considerably different from the one of the upper state, the system can exhibit population inversion for some values of the phase around Φ = 2nπ. We have interpreted and discussed this result in terms of coherent population trapping. Nevertheless, in all regimes here considered the population of the upper state vanishes for Φ = (2n+ 1)π. We have explained these behaviors using a convenient basis, showing that the dynamics is given by the concurrence of the Hamiltonian evolution, which is phase-sensitive, with the structure and non-unitarity of the relaxation processes. In particular, for Φ = (2n+ 1)π
the steady state of the system corresponds to the steady state of a (closed) two-level transition.
The phase dependence of the Hamiltonian evolution in closed-loop schemes shares many analogies with an atom interferometer [3]. The phase dependence survives also at steady state [4, 5], and the response of the system could be used as a device for measuring the relative phase between laser fields. For instance, in the♦system the phase could be measured through the population of the upper state. In fact, for sufficiently-weak fields, the functional behavior of this population is well approximated by cos2Φ/2, and the fluorescence signal from the upper state shows the features
of an interference pattern which is sensitive to Φ.
This study may be useful in the spectroscopy of alkali-earth atoms, currently investigated in experiments aiming at optical frequency standards [10] or at Bose-Einstein condensation by all-optical means [11]. Further, efficient laser-cooling schemes for these kind of atoms could be developed, by exploiting the phase properties due to the atomic motion in proper laser geometries [17].
Finally, the♦scheme exemplifies a system where non-linear optics with resonant atoms can be realized. Here, the phase is a control parameter capable to change the response of the medium to the drive [5, 6, 7, 8, 18, 19]. This will be object of future investigations.
ACKNOWLEDGMENTS
and QUANTIM, EPSRC (GR/R04096) and the Leverhulme Trust. GLO acknowledges support from SGI. ∗ Present address: University of Ulm, Abteilung Quantenphysik, Albert-Einstein-Allee 11, D-89081 Ulm, Germany.
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[12] A systematic investigation should include the mechanical effects of light on the center-of-mass motion dynamics, and the OBE should describe the coupled dynamics of internal and external degrees of freedom. Nevertheless, in the limit in which the internal relaxation is much faster than the rate with which the atomic center-of-mass coordinates evolve on a macroscopic scale, one can evaluate the internal steady state, treating these coordinates as fixed parameters. Our analysis is restricted to this short-time behavior.
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[14] Note that for Φ = (2n+ 1)πpopulation inversion has been predicted in the double-Λ configuration for different decay rates from the excited states [4].
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FIG. 1: Atomic level scheme and nomenclature or the relevant levels for the♦configuration. The inset displays the Double-Λ system for comparison.
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